Normal map texture does not render on all faces
(((While preparing this question by reproducing my problem in a clean Blender file, to my surprise the problem did not exist. By slowly rebuilding my original scene, I found where the problem comes from, hence I am posting this self-answer.)))

The problem re-appeared as soon as I changed the lighting to "Sun" and turned on "Environment Lighting" to lit the parts not directly lit by the sun. As a consequence, the faces that were not directly lit by the sun did not render their normal maps applied to them.

Therefore, it seems that "Environment Lighting" does not cast rays. Then, if rays do not hit the face, there are no rays to be reflected in different directions depending on the precise location that the rays are incident (which is what the normal map does), and therefore the "normal map will not render".

Now, to have lighting on faces not lit directly by the sun, "Environment Lighting" may be replaced by adding a "Hemi" lamp. This lamp behaves the same as "Environment Lighting" (i.e., light on all directions), but apparently does cast rays, allowing the normal map to be rendered correctly:



Still unsolved questions:


What happened when the sun was changed to "No Shadow" in the OP?
Why does Environmental Lighting work the way it does? Is there a particular reason for which that would be the desired behaviour?